We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. Here, we restrict ourselves to pseudodifferential operators with x-independent symbols (Fourier multipliers). We show that a parabolic toroidal pseudodifferential operator generates an analytic semigroup on the Besov space Bspq(Tn,E) and on the Sobolev space Wkp(Tn,E), where E is an arbitrary Banach space, 1≤p,q≤∞, s∈R and k∈N0. For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.

Barraza Martínez, Bienvenido2016-03-18T14:40:18ZHernández Monzón, JairoDenk, Robert2016-03-18T14:40:18Z2016-08Hernández Monzón, JairoWe consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. Here, we restrict ourselves to pseudodifferential operators with x-independent symbols (Fourier multipliers). We show that a parabolic toroidal pseudodifferential operator generates an analytic semigroup on the Besov space B<sup>s</sup><sub>pq</sub>(T<sup>n</sup>,E) and on the Sobolev space W<sup>k</sup><sub>p</sub>(T<sup>n</sup>,E), where E is an arbitrary Banach space, 1≤p,q≤∞, s∈R and k∈N<sub>0</sub>. For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the TorusBarraza Martínez, BienvenidoNau, TobiasengNau, TobiasDenk, Robert